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Treat a small patch as the point tensile-crack seismic source at ${\mathbf{y}}_0$, with $M(t)=KV(t)$ and $K=\lambda I+2\mu \mathbf{n}{\mathbf{n}}^{\mathsf{T}}$. Let $\mathbf{e}=(\mathbf{x}-{\mathbf{y}}_0)/R$, $\gamma =\mathbf{e}\cdot \mathbf{n}$, $\mathbf{a}=K\mathbf{e}$, $m=\mathbf{e}\cdot K\mathbf{e}$ and $s=\mathrm{tr}K$. Source convolution and an integration by parts give the point-moment elastodynamic displacementThe derivatives act on angular factors and on retarded times as well as on powers of $R$; keeping only the explicit $1/R$ Green-function terms would miss the static displacement.

Set $V_P=V(t-R/\alpha )$, $V_S=V(t-R/\beta )$ andConvolving the elastodynamic Green tensor with the opening history givesUse ${\partial }_{j}R=e_j$, ${\partial }_j{e}_i=({\delta }_{ij}-e_i{e}_j)/R$, ${\partial }_j{V}_P=-e_j{\dot{V}}_P/\alpha$, andFor example, $K_{kj}{\partial }_j(e_i{e}_k/R)=[a_i+e_i(s-3m)]/R^2$, while differentiating the near-field angular tensor gives $[6a_i+3e_{i}s-15e_{i}m]/R^4$. Combining these with the endpoint terms in ${\partial }_{R}I$ yieldsFor the normal crack, $\mathbf{a}=\lambda \mathbf{e}+2\mu \gamma \mathbf{n}$, $m=\lambda +2\mu {\gamma }^2$, and $s=3\lambda +2\mu$. Explicitly, the three nonradiative vector coefficients in the last two lines areThus those lines are ${\mathbf{C}}_P{V}_P/({\alpha }^2{R}^2)+{\mathbf{C}}_S{V}_S/({\beta }^2{R}^2)+{\mathbf{C}}_{I}I/R^4$. The small-source approximation assumes the patch size is small relative to both $R$ and the wavelengths of interest; a finite crack is obtained by integrating the same point-patch expression over its surface with its local opening and normal.

Let $d(\mathbf{y},t)=[u^{[n]}]$ denote the signed opening, positive when the faces separate, and let $S$ be the crack surface. Its displacement jump is $\mathbf{b}=d\mathbf{n}$. The singular strain associated with this jump has moment density $c_{ijkl}{b}_k{n}_l$. For the isotropic elastic stiffness tensor, contraction givesThe equivalent distributional body force is the negative divergence of this prescribed moment density:The resulting forced elastic wave in an isotropic solid equation is $\rho u_{i,tt}={\partial }_j{\sigma }_{ij}+f_i$. This is a tensile-crack seismic source; it need not be represented as an ordinary smooth volume force. For a small nearly planar patch of area $S_0$, uniform normal $\mathbf{n}$ and opening $d(t)$, define its volume change $V(t)=S_{0}d(t)$. Its seismic moment tensor and point-source force areThe sign convention makes a positive opening a positive extensional moment. Reversing the convention for the displacement jump changes both signs together; the force is independent of reversing the orientation of the crack normal.

For a planar tensile-crack seismic source, $\theta$ is measured from its normal. The P wave displacement is radial with the first displayed amplitude. The S wave displacement is tangential in the plane containing the observation direction and crack normal, with four alternating signed lobes and nodes at the poles and equator. Both radiative fields depend on the opening rate and vanish once opening stops; the static displacement field depends on the final opening.